I´m glad you used the example of The Barrow-wights, since I consider examples important. A Barrow-wight is a specific kind of wights with different connotations than just wights in general, which Magpie´s etymological post shows (in which wight also was used about unpleasant creatures that are hardly more uncanny than gnomes). Both from the quote you´ve given and from what I can glean from different traditions and notable stories about barrow wights, it is associated with spirits animating the remains of a barrow, so here I agree that we have an example which seems clearly open to the kind of interpretation that we´ve discussed.

The story of LoTR in itself also says several concrete things about these wights, giving the wight the ability to perform spells and connecting their spells to objects within the barrow.

I can´t say anything about requirements to be undead, as I haven´t got any clear idea about what the term means. As to the nature of and the powers of the Nazûl, they are explained by Gandalf and Aragorn in LoTR and elaborated on by Tolkien in his letters, and it´s made clear that they are controlled by Sauron and are extensions of his malice (unlike more independent evil creatures).